Limits - Formal Definitions

NOTE:  You won't be responsible for this material in this class.   However, I still encourage all of you to read this over carefully.

We say that "the limit as x approaches a of f(x)  is L", written

                                                          

provided that for any e > 0, there exists a d > 0 such that for all x values within d of a (but not equal to a), it is true that f(x) is within e of L.     

(  e is the Greek letter epsilon, and d is the Greek letter delta. )

More simply put, if x is very close to a then f(x) is very close to L.

There is a useful mathematical short-hand used by mathematicians around the world.  Some useful symbols are:    

“for all”  =  

“there exists” = 

“such that” =

“A implies B” means the same thing as “if A then B”, and is written:

                                                                

Using this notation, the above definition of limit can be more compactly written:

                  

This says that no matter how small you choose e to be, it is possible to find a corresponding d which guarantees that whenever x is within a distance of d from a (but not equal to a), we will be certain that f(x) is within a distance of e from L.   

The formal limit definition often seems at first strange to students, and indeed much calculus evolved without any need of it.   Eventually though, the formal definition was needed to put calculus on a solid logical foundation.   Vague intuitive notions of limits can only take one so far;  they needed ultimately to be wedded to precise definitions before calculus could be successfully applied to non-intuitive fields like quantum mechanics, and to other areas of mathematics.

Take a look at the proofs of the basic limit properties.  From the above formal definition, we'll derive various useful limit properties.

There are other ways to construct the foundations of calculus.  But enough for now!